Weighted composition operators on weighted Dirichlet spaces: boundedness, compactness and spectral properties
Anirban Sen
TL;DR
This paper studies weighted composition operators $C_{\psi,\varphi}$ on weighted Dirichlet spaces $\mathcal{D}_{\alpha}$ for $\alpha\in(-1,1)$. It derives verifiable necessary and sufficient conditions for boundedness and compactness in terms of the inducing functions $\psi$ and $\varphi$, including a precise characterization when $\alpha\in(0,1)$ and a vanishing-condition criterion for compactness on $\mathcal{D}_{\alpha}$. It also proves a weighted Comparison Theorem across the Dirichlet scale and provides explicit spectral descriptions for operators with fixed-point inducing maps, showing that nonzero spectrum is generated by $\psi(a)(\varphi'(a))^n$ and that, under compactness, the spectrum consists of these values together with $0$. The results extend classical unweighted Dirichlet- and Hardy-space theory to weighted settings, offering practical criteria and exact spectra for a broad class of weighted composition operators with concrete examples.
Abstract
We establish necessary and sufficient conditions for the boundedness and compactness of weighted composition operators acting on weighted Dirichlet spaces and determine the spectrum of a certain class of such operators. Our results extend earlier work on unweighted composition operators and highlight the close interplay between the operator theoretic behavior of weighted composition operators and the function theoretic properties of their inducing functions. Several examples are provided to illustrate the applicability of the obtained results.